The group-theoretical classification of the 11-dimensional classical homogeneous Kaluza–Klein cosmologies

Abstract

In the context of the classical Kaluza–Klein cosmology the generalized Bianchi models in 11 dimensions are considered. These are space-times whose spacelike ten-dimensional sections are the hypersurfaces of transitivity for a ten-dimensional isometry group of the total space-time. Such a space-time is a trivial principal fiber bundle P(M,G7), where M is a four-dimensional physical space-time with an isometry group G3 (of a Bianchi type) and G7 is a compact isometry group of the compact internal space. The isometry group of P is G10=G3⊗G7, hence all the generalized Bianchi models are classified by enumerating the relevant groups G7. Due to the compactness of G7 the result is astonishingly simple: there are three distinct homogeneous internal spaces in addition to the 11 ordinary Bianchi types for M.

Non-semisimple Lie algebras with Levi factor o (3), (2, ) and their invariants

“…In particular, the analysis undertaken has given a representation theoretic interpretation of the invariants obtained for the (3 + 1) kinematical algebras like the Galilei algebra. The interest of these Levi factors is therefore justified not only by kinematical problems, but also by the extensions of Bianchi type-IX cosmology [20,21]. Specially interesting are those admissible extensions which have no invariants.…”

“…The results obtained so far for the Levi factors so (3) and sl (2, R) have important physical applications, like the classification of multidimensional spacetimes [21]. In this frame, all ten dimensional real Lie algebras having a (7 + d)dimensional compact subalgebra have been determined.…”

“…The algebra s − → ⊕ R r is of interest for multidimensional extensions of the Bianchi type-IX cosmology [20], and the corresponding vacuum Einstein field equations have been solved in [20]. Indeed this is the simplest embedding of a Bianchi type-IX algebra in an algebra with non-trivial Levi decomposition [21]. It can easily be verified that…”

Hidden Symmetries in a Mixmaster-Type Universe

“…Affine algebras proved useful in describing various interesting physical problems [15]. Cartan matrix 1 of the mixmaster model (9). There are three nodes and connected them double lines.…”

“…× M n describing Kasner-like behavior near a singularity. Full classification of homogeneous multidimensional Kaluza -Klein models is presented in [9,10]. Models have been investigated by numerical methods [11] and explicit integration [12].…”